International Journal of Algebra and Computation

We study properties of automorphisms of graph products of groups. We show that graph product $\mathrm{\Gamma }{𝒢}$ has nontrivial pointwise inner automorphisms if and only if some vertex group corresponding to a central vertex has nontrivial pointwise inner automorphisms. We use this result to study residual finiteness of $\mathrm{\text{Out}}(\mathrm{\Gamma }{𝒢})$. We show that if all vertex groups are finitely generated residually finite and the vertex groups corresponding to central vertices satisfy certain technical (yet natural) condition, then $\mathrm{\text{Out}}(\mathrm{\Gamma }{𝒢})$ is residually finite. Finally, we generalize this result to graph products of residually $p$-finite groups to show that if $\mathrm{\Gamma }{𝒢}$ is a graph product of finitely generated residually $p$-finite groups such that the vertex groups corresponding to central vertices satisfy the $p$-version of the technical condition then $\mathrm{\text{Out}}(\mathrm{\Gamma }{𝒢})$ is virtually residually $p$-finite. We use this result to prove bi-orderability of Torreli groups of some graph products of finitely generated residually torsion-free nilpotent groups.

We describe an algorithm which, given an automorphism $\phi$ of a free group $F$ of finite rank, computes a basis of the fixed point subgroup $\mathrm{\text{Fix}}(\phi )$.

We give a characterization of isomorphisms between Schreier graphs in terms of the groups, subgroups and generating systems. This characterization may be thought as a graph analog of Mostow’s rigidity theorem for hyperbolic manifolds. This allows us to give a transitivity criterion for Schreier graphs. Finally, we show that Tarski monsters satisfy a strong simplicity criterion. This gives a partial answer to a question of Benjamini and Duminil-Copin.

Let ${𝒞}\subset {\mathbb{P}}^r$ be a nondegenerate projective integral curve of degree $r+1$ which is not linearly normal. In this paper, we continues the study begun in [E. Park, Projective curves of degree=codimension+2, Math. Z. 256 (2007) 685–697] for the minimal free resolution of ${𝒞}$. It is well-known that ${𝒞}$ is an isomorphic projection of a rational normal curve $\widetilde{{𝒞}}\subset {\mathbb{P}}^{r+1}$ from a point $P\in {\mathbb{P}}^{r+1}$. Our main result is about how the graded Betti numbers of ${𝒞}$ are determined by the rank of $P$ with respect to $\widetilde{{𝒞}}$, which is a measure of the relative location of $P$ from $\widetilde{{𝒞}}$.

We show that the class of finitely generated virtually free groups is precisely the class of finitely generated demonstrable subgroups for Thompson’s group $V$. The class of demonstrable groups for $V$ consists of all groups which can embed into $V$ with a natural dynamical behavior in their induced actions on the Cantor space ${ℭ}_2:={\{0,1\}}^{\omega }$. There are also connections with formal language theory, as the class of groups with context-free word problem is also the class of finitely generated virtually free groups, while Thompson’s group $V$ is a candidate as a universal $co{𝒞}{ℱ}$ group by Lehnert’s conjecture, corresponding to the class of groups with context free co-word problem (as introduced by Holt, Rees, Röver, and Thomas). Our main results answers a question of Berns-Zieve, Fry, Gillings, Hoganson, and Matthews, and separately of Bleak and Salazar-Díaz, and it fits into the larger exploration of the class of $co{𝒞}{ℱ}$ groups as it shows that all four of the known closure properties of the class of $co{𝒞}{ℱ}$ groups hold for the set of finitely generated subgroups of $V$.

For $n$ a positive integer and $K$ a finite set of finite algebras, let $L(n,K)$ denote the largest $n$-generated subdirect product whose subdirect factors are algebras in $K$. When $K$ is the set of all $n$-generated subdirectly irreducible algebras in a locally finite variety ${𝒱}$, then $L(n,K)$ is the free algebra $F_{{𝒱}}(n)$ on $n$ free generators for ${𝒱}$. For a finite algebra $A$ the algebra $L(n,\{A\})$ is the largest $n$-generated subdirect power of $A$. For every $n$ and finite $A$ we provide an upper bound on the cardinality of $L(n,\{A\})$. This upper bound depends only on $n$ and these basic parameters: the cardinality of the automorphism group of $A$, the cardinalities of the subalgebras of $A$, and the cardinalities of the equivalence classes of certain equivalence relations arising from congruence relations of $A$. Using this upper bound on $n$-generated subdirect powers of $A$, as $A$ ranges over the $n$-generated subdirectly irreducible algebras in ${𝒱}$, we obtain an upper bound on $|F_{{𝒱}}(n)|$. And if all the $n$-generated subdirectly irreducible algebras in ${𝒱}$ have congruence lattices that are chains, then we characterize in several ways those ${𝒱}$ for which this upper bound is obtained.

Let $Y$ be a smooth hypersurface of degree $d$ in a projective space ${\mathbb{P}}^n$ and take a point $y$ in ${\mathbb{P}}^n\backslash Y$. Let ${\mathrm{\text{Hilb}}}_y^{[m]}(Y/{\mathbb{P}}^{n-1})$ be the relative Hilbert scheme parametrizing zero-dimensional subscheme, of length $m$, of fibers of the projection morphism $Y\to {\mathbb{P}}^{n-1}$ from $y$. In this paper we present an embedding of the relative Hilbert scheme ${\mathrm{\text{Hilb}}}_y^{[m]}(Y/{\mathbb{P}}^{n-1})$ into a weighted projective space and describe its defining ideal for general $y$. We also study line bundles on the relative Hilbert scheme ${\mathrm{\text{Hilb}}}_y^{[m]}(Y/{\mathbb{P}}^{n-1})$ for $n\ge 4$ and general $y$.

The class of finitely presented algebras over a field $K$ with a set of generators $a_1,\dots ,a_n$ and defined by homogeneous relations of the form $a_1{a}_2\cdots a_n=a_{\sigma (1)}{a}_{\sigma (2)}\cdots a_{\sigma (n)}$, where $\sigma$ runs through a subset $H$ of the symmetric group ${Sym}_n$ of degree $n$, is investigated. Groups $H$ in which the cyclic group $〈(1,2,\dots ,n)〉$ is a normal subgroup of index $2$ are considered. Certain representations by permutations of the dihedral and semidihedral groups belong to this class of groups. A normal form for the elements of the underlying monoid $S_n(H)$ with the same presentation as the algebra is obtained. Properties of the algebra are derived, it follows that it is an automaton algebra in the sense of Ufnarovskij. The universal group $G_n$ of $S_n(H)$ is a unique product group, and it is the central localization of a cancellative subsemigroup of $S_n(H)$. This, together with previously obtained results on such semigroups and algebras, is used to show that the algebra $K[S_n(H)]$ is semiprimitive.